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Algebras and modules in monoidal model categories. (English) Zbl 1026.18004

Introduction: In recent years the theory of structured ring spectra (formerly known as \(A_\infty\)- and \(E_\infty\)-ring spectra) has been significantly simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra. In order to make use of all of the standard tools from homotopy theory, it is important to have a Quillen model category structure [D. G. Quillen, “Homotopical algebra”, Lect. Notes Math. 43 (1967; Zbl 0168.20903)] available here. In this paper we provide a general method for lifting model structures to categories of rings, algebras, and modules. This includes, but is not limited to, each of the new theories of ring spectra.
One model for structured ring spectra is given by the \(S\)-algebras of A. D. Elmendorf, T. Kríz, M. A. Mandell and J. P. May [“Rings, modules, and algebras in stable homotopy theory”, Math. Surv. Monogr. 47 (1997; Zbl 0894.55001)]. This example has the special feature that every object is fibrant, which makes it easier to form model structures of modules and algebras. There are other new theories such as ‘symmetric ring spectra’ [M. Hovey, B. Shipley and J. Smith, J. Am. Math. Soc. 13, 149-208 (2000; Zbl 0931.55006)], ‘functors with smash product’ [M. Bökstedt, W. C. Hsiang and I. Madsen, Invent. Math. 111, 465-539 (1993; Zbl 0804.55004)] or ‘diagram ring spectra’ [M. A. Mandell, J. P. May, S. Schwede and B. Shipley, Proc. Lond. Math. Soc., III. Ser. 82, 441-512 (2001; Zbl 1017.55004)] which do not have this special property. This paper provides the necessary input for obtaining model categories of associative structured ring spectra in these contexts. Categories of commutative ring spectra appear to be intrinsically more complicated, and they are not treated systematically here. Our general construction of model structures for associative monoids also gives a unified treatment of previously known cases (simplicial sets, simplicial abelian groups, chain complexes, \(S\)-modules) and applies to other new examples (\(\Gamma\)-spaces and modules over group algebras). We discuss these examples in more detail in §5.
Technically, what we mean by an ‘algebra’ is a monoid in a symmetric monoidal category, for example, a ring in the category of abelian groups under tensor product. To work with this symmetric monoidal product it must be compatible with the model category structure, which leads to the definition of a monoidal model category. To obtain a model category structure of algebras we have to introduce one further axiom, the monoid axiom. A filtration on certain pushouts of monoids is then used to reduce the problem to standard model category arguments based on Quillen’s ‘small object argument’. The case of modules also uses the monoid axiom, but the argument here is straightforward. Our main result is stated in Theorem 4.1:
Let \({\mathcal C}\) be a cofibrantly generated, monoidal model category. Assume further that every object in \({\mathcal C}\) is small relative to the whole category and that \({\mathcal C}\) satisfies the monoid axiom.
(1) Let \(R\) be a monoid in \({\mathcal C}\). Then the category of left \(R\)-modules is a cofibrantly generated model category.
(2) Let \(R\) be a commutative monoid in \({\mathcal C}\). Then the category of \(R\)-modules is a cofibrantly generated, monoidal model category satisfying the monoid axiom.
(3) Let \(R\) be a commutative monoid in \({\mathcal C}\). Then the category of \(R\)-algebras is a cofibrantly generated model category. Every cofibration of \(R\)-algebras whose source is cofibrant as an \(R\)-module is also a cofibration of \(R\)-modules. In particular, if the unit \(\mathbb{I}\) of the smash product is cofibrant in \({\mathcal C}\), then every cofibrant \(R\)-algebra is also cofibrant as an \(R\)-module.

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
18G55 Nonabelian homotopical algebra (MSC2010)
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